I.INTRODUCTION

An eight sentence history of Newtonian mechanics1 shows how muchthe subject has developed since Newton introduced F = dp/dt in thesecond half of the 1600s.2 In the mid-1700s Euler devisedand applied a version of the principle of least actionusing mostly geometrical methods. In 1755 the 19-year-old Joseph-Louis Lagrangesent Euler a letter that streamlined Euler's methods into algebraicform. "[A]fter seeing Lagrange's work Euler dropped his own method,espoused that of Lagrange, and renamed the subject the calculusof variations."3 Lagrange, in his 1788 Analytical Mechanics,4 introduced whatwe call the Lagrangian function and Lagrange's equations of motion.About half a century later (1834–1835) Hamilton published Hamilton's principle,5to which Landau and Lifshitz6 and Feynman7 reassigned the nameprinciple of least action.8 Between 1840 and 1860 conservation ofenergy was established in all its generality.9 In 1918 Noether10proved several relations between symmetries and conserved quantities. In the1940s Feynman11 devised a formulation of quantum mechanics that notonly explicitly underpins the principle of least action, but alsoshows the limits of validity of Newtonian mechanics.

Except forconservation of energy, students in introductory physics are typically introducedto the mechanics of the late 1600s. To modernize thistreatment, we have suggested12 that the principle of least actionand Lagrange's equations become the basis of introductory Newtonian mechanics.Recent articles discuss how to use elementary calculus to deriveNewton's laws of motion,13 Lagrange's equations,14 and examples of Noether'stheorem15 from the principle of least action, describe the modernrebirth of Euler's methods16 and suggest ways in which upperundergraduate physics classes can be transformed using the principle ofleast action.17

How are these concepts and methods to beintroduced to undergraduate physics students? In this paper we suggesta reversal of the historical order: Begin with conservation ofenergy and graduate to the principle of least action andLagrange's equations. The mathematical prerequisites for the proposed course includeelementary trigonometry, polar coordinates, introductory differential calculus, partial derivatives, andthe idea of the integral as a sum of increments.

The story line presented in this article omits most detailsand is offered for discussion, correction, and elaboration. We donot believe that a clear story line guarantees student understanding.On the contrary, we anticipate that trials of this approachwill open up new fields of physics education research.

II. ONE-DIMENSIONAL MOTION: ANALYTICSOLUTIONS

We start by using conservation of energy to analyze particlemotion in a one-dimensional potential. Much of the power ofthe principle of least action and its logical offspring, Lagrange'sequations, results from the fact that they are based onenergy, a scalar. When we start with conservation of energy,we not only preview more advanced concepts and procedures, butalso invoke some of their power. For example, expressions forthe energy which are consistent with any constraints automatically eliminatethe corresponding constraint forces from the equations of motion. Byusing the constraints and constants of the motion, we oftencan reduce the description of multi-dimensional systems to one coordinate,whose motion can then be found using conservation of energy.Equilibrium and statics also derive from conservation of energy.

Wefirst consider one-dimensional motion in a uniform vertical gravitational field.Heuristic arguments lead to the expression mgy for the potentialenergy. We observe, with Galileo, that the velocity of aparticle in free fall from rest starting at position y = 0decreases linearly with time: = –gt, where we have expressed thetime derivative by a dot over the variable. This relationintegrates to the form

We multiply Eq. (1) by mgand rearrange terms to obtain the first example of conservationof energy:

where the symbol E represents the total energyand the symbols K and U represent the kinetic andpotential energy, respectively.

A complete description of the motion ofa particle in a general conservative one-dimensional potential follows fromthe conservation of energy. Unfortunately, an explicit function of theposition versus time can be derived for only a fractionof such systems. Students should be encouraged to guess analyticsolutions, a powerful general strategy because any proposed solution iseasily checked by substitution into the energy equation. Heuristic guessesare assisted by the fact that the first time derivativeof the position, not the second, appears in the energyconservation equation.

The following example illustrates the guessing strategy forlinear motion in a parabolic potential. This example also introducesthe potential energy diagram, a central feature of our storyline and an important tool in almost every undergraduate subject.

A. Harmonic oscillator

Our analysis begins with a qualitative prediction ofthe motion of a particle in a parabolic potential (orin any potential with motion bounded near a single potentialenergy minimum). If we consider the potential energy diagram fora fixed total energy, we can predict that the motionwill be periodic. For a parabolic potential (Fig. 1) conservationof energy is expressed as

Weset = t and equate the right-hand sides of Eqs. (4)and (5) and obtain the solution

If we take thetime derivative of x in Eq. (6) and use itto equate the left-hand sides of Eqs. (4) and (5),we find that18

The fact that does not dependon the total energy E of the particle means thatthe period is independent of E and hence independent ofthe amplitude of oscillation described by Eq. (6). The simpleharmonic oscillator is widely applied because many potential energy curvescan be approximated as parabolas near their minima.

At thispoint, it would be desirable to introduce the concept ofthe worldline, a position versus time plot that completely describesthe motion of a particle.

III. ACCELERATION AND FORCE

Inthe absence of dissipation, the force can be defined interms of the energy. We start with conservation of energy:

We take the time derivative of both sides and usethe chain rule:

By invoking the tendency of a ballto roll downhill, we can define force as the negativespatial derivative of the potential,

from which we see thatF = –kx for the harmonic potential and F = –mg for the gravitationalforce near the earth's surface.

IV. NONINTEGRABLEMOTIONS IN ONE DIMENSION

We guessed a solution for simple harmonicmotion, but it is important for students to know thatfor most mechanical systems, analytical solutions do not exist, evenwhen the potential can be expressed analytically. For these cases,our strategy begins by asking students to make a detailedqualitative prediction of the motion using the potential energy diagram,for example, describing the velocity, acceleration, and force at differentparticle positions, such as A through F in Fig. 1.

The next step might be to ask the student toplot by hand a few sequential points along the worldlineusing a difference equation derived from energy conservation corresponding toEq. (3),

where U(x) describes an arbitrary potential energy. Theprocess of plotting necessarily invokes the need to specify theinitial conditions, raises the question of accuracy as a functionof step size, and forces an examination of the behaviorof the solution at the turning points. Drawing the resultingworldline can be automated using a spreadsheet with graphing capabilities,perhaps comparing the resulting approximate curve with the analytic solutionfor the simple harmonic oscillator.

After the drudgery of thesepreliminaries, students will welcome a more polished interactive display thatnumerically integrates the particle motion in a given potential. Onthe potential energy diagram (see Fig. 2), the student setsup initial conditions by dragging the horizontal energy line upor down and the particle position left or right. Thecomputer then moves the particle back and forth along theE-line at a rate proportional to that at which theparticle will move, while simultaneously drawing the worldline (upper plotof Fig. 2).

V. REDUCTION TO ONE COORDINATE

The analysis ofone-dimensional motion using conservation of energy is powerful but limited.In some important cases we can use constraints and constantsof the motion to reduce the description to one coordinate.In these cases we apply our standard procedure: qualitative analysisusing the (effective) potential energy diagram followed by interactive computersolutions. We illustrate this procedure by some examples.

A. Projectilemotion

For projectile motion in a vertical plane subject to auniform vertical gravitational field in the y-direction, the total energyis

The potential energy is not a function of x;therefore, as shown in the following, momentum in the x-direction,px, is a constant of the motion. The energy equationbecomes

In Newtonian mechanics the zero of the energy isarbitrary, so we can reduce the energy to a singledimension y by making the substitution:

Our analysis of projectilemotion already has applied a limited version of a powerfultheorem due to Noether.19 The version of Noether's theorem usedhere says that when the total energy E is notan explicit function of an independent coordinate, x for example,then the function E/ is a constant of the motion.20We have developed a simple, intuitive derivation of this versionof Noether's theorem. The derivation is not included in thisbrief story line.

The above strategy uses a conservation lawto reduce the number of dimensions. The following example usesconstraints to the same end.

B. Object rolling without slipping

Two-dimensionalcircular motion and the resulting kinetic energy are conveniently describedusing polar coordinates. The fact that the kinetic energy isan additive scalar leads quickly to its expression in termsof the moment of inertia of a rotating rigid body.When the rotating body is symmetric about an axis ofrotation and moves perpendicularly to this axis, the total kineticenergy is equal to the sum of the energy ofrotation plus the energy of translation of the center ofmass. (This conclusion rests on the addition of vector components,but does not require the parallel axis theorem.) Rolling withoutslipping is a more realistic idealization than sliding without friction.

We use these results to reduce to one dimension thedescription of a marble rolling along a curved ramp thatlies in the x-y plane in a uniform gravitational field.Let the marble have mass m, radius r, and momentof inertia Imarble. The nonslip constraint tells us that v = r.Conservation of energy leads to the expression:

where

The motionis described by the single coordinate y. Constraints are usedtwice in this example: explicitly in rolling without slipping andimplicitly in the relation between the height y and thedisplacement along the curve.

C. Motion in a central gravitationalfield

We analyze satellite motion in a central inverse-square gravitational fieldby choosing the polar coordinates r and in theplane of the orbit. The expression for the total energyis

The angle does not appear in Eq. (17).Therefore we expect that a constant of the motion isgiven by E/, which represents the angular momentum J. (Wereserve the standard symbol L for the Lagrangian, introduced laterin this paper.)

Students employ the plot ofthe effective potential energy Ueff(r) to do the usual qualitativeanalysis of radial motion followed by computer integration. We emphasizethe distinctions between bound and unbound orbits. With additional useof Eq. (18), the computer can be programmed to plota trajectory in the plane for each of these cases.

D. Marble, ramp, and turntable

This system is more complicated, butis easily analyzed by our energy-based method and more difficultto treat using F = ma. A relatively massive marble of massm and moment of inertia Imarble rolls without slipping alonga slot on an inclined ramp fixed rigidly to alight turntable which rotates freely so that its angular velocityis not necessarily constant (see Fig. 3). The moment ofinertia of the combined turntable and ramp is Irot. Weassume that the marble stays on the ramp and findits position as a function of time.

We start witha qualitative analysis. Suppose that initially the turntable rotates andthe marble starts at rest with respect to the ramp.If the marble then begins to roll up the ramp,the potential energy of the system increases, as does thekinetic energy of the marble due to its rotation aroundthe center of the turntable. To conserve energy, the rotationof the turntable must decrease. If instead the marble beginsto roll down the ramp, the potential energy decreases, thekinetic energy of the marble due to its rotation aroundthe turntable axis decreases, and the rotation rate of theturntable will increase to compensate. For a given initial rotationrate of the turntable, there may be an equilibrium valueat which the marble will remain at rest. If themarble starts out displaced from this value, it will oscillateback and forth along the ramp.

More quantitatively, the squareof the velocity of the marble is

Conservation of energyyields the relation

where M indicates the marble's mass augmentedby the energy effects of its rolling along the ramp,Eq. (16). The right side of Eq. (21) is notan explicit function of the angle of rotation . Thereforeour version of Noether's theorem tells us that E/ isa constant of the motion, which we recognize as thetotal angular momentum J:

If we substitute into Eq. (21)the expression for from Eq. (22), we obtain

Thevalues of E and J are determined by the initialconditions. The effective potential energy Ueff(x) may have a minimumas a function of x, which can result in oscillatorymotion of the marble along the ramp. If the marblestarts at rest with respect to the ramp at theposition of minimum effective potential, it will not move alongthe ramp, but execute a circle around the center ofthe turntable.

VI. EQUILIBRIUM AND STATICS: PRINCIPLE OF LEAST POTENTIALENERGY

Our truncated story line does not include frictional forces oran analysis of the tendency of systems toward increased entropy.Nevertheless, it is common experience that motion usually slows downand stops. Our use of potential energy diagrams makes straightforwardthe intuitive formulation of stopping as a tendency of asystem to reach equilibrium at a local minimum of thepotential energy. This result can be formulated as the principleof least potential energy for systems in equilibrium. Even inthe absence of friction, a particle placed at rest ata point of zero slope in the potential energy curvewill remain at rest. (Proof: An infinitesimal displacement results inzero change in the potential energy. Due to energy conservation,the change in the kinetic energy must also remain zero.Because the particle is initially at rest, the zero changein kinetic energy forbids displacement.) Equilibrium is a result ofconservation of energy.

Here, as usual, Feynman is ahead ofus. Figure 4 shows an example from his treatment ofstatics.21 The problem is to find the value of thehanging weight W that keeps the structure at rest, assuminga beam of negligible weight. Feynman balances the decrease inthe potential energy when the weight W drops 4 withincreases in the potential energy for the corresponding 2 riseof the 60 lb weight and the 1 rise inthe 100 lb weight. He requires that the net potentialenergy change of the system be zero, which yields

orW = 55 lb. Feynman calls this method the principle of virtual work,which in this case is equivalent to the principle ofleast potential energy, both of which express conservation of energy.

There are many examples of the principle of minimum potentialenergy, including a mass hanging on a spring, a lever,hydrostatic balance, a uniform chain suspended at both ends, avertically hanging slinky,22 and a ball perched on top ofa large sphere.

VII.FREE PARTICLE. PRINCIPLE OF LEAST AVERAGE KINETIC ENERGY

For all itspower, conservation of energy can predict the motion of onlya fraction of mechanical systems. In this and Sec. VIIIwe seek a principle that is more fundamental than conservationof energy. One test of such a principle is thatit leads to conservation of energy. Our investigations will employtrial worldlines that are not necessarily consistent with conservation ofenergy.

We first think of a free particle initially atrest in a region of zero potential energy. Conservation ofenergy tells us that this particle will remain at restwith zero average kinetic energy. Any departure from rest, sayby moving back and forth, will increase the average ofits kinetic energy from the zero value. The actual motionof this free particle gives the least average kinetic energy.The result illustrates what we will call the principle ofleast average kinetic energy.

Now we view the same particlefrom a reference frame moving in the negative x-direction withuniform speed. In this frame the particle moves along astraight worldline. Does this worldline also satisfy the principle ofleast average kinetic energy? Of course, but we can checkthis expectation and introduce a powerful graphical method established byEuler.16

We fix two events A and C at theends of the worldline (see Fig. 5) and vary thetime of the central event B so that the kineticenergy is not the same on the legs labeled 1and 2. Then the time average of the kinetic energyK along the two segments 1 and 2 of theworldline is given by

We multiply both sides of Eq.(25) by the fixed total time and recast the velocityexpressions using the notation in Fig. 5:

We find theminimum value of the average kinetic energy by taking thederivative with respect to the time t of the centralevent:

As expected,when the time-averaged kinetic energy has a minimum value, thekinetic energy for a free particle is the same onboth segments; the worldline is straight between the fixed initialand final events A and C and satisfies the principleof minimum average kinetic energy. The same result follows ifthe potential energy is uniform in the region under consideration,because the uniform potential energy cannot affect the kinetic energyas the location of point B changes on the spacetimediagram.

In summary, we have illustrated the fact that forthe special case of a particle moving in a regionof zero (or uniform) potential energy, the kinetic energy isconserved if we require that the time average of thekinetic energy has a minimum value. The general expression forthis average is

In an introductory text we might introduceat this point a sidebar on Fermat's principle of leasttime for the propagation of light rays.

Now we areready to develop a similar but more general law thatpredicts every central feature of mechanics.

VIII. PRINCIPLE OFLEAST ACTION

Section VI discussed the principle of least potential energyand Sec. VII examined the principle of least average kineticenergy. Along the way we mentioned Fermat's principle of leasttime for ray optics. We now move on to theprinciple of least action, which combines and generalizes the principlesof least potential energy and least average kinetic energy.

A.Qualitative demonstration

We might guess (incorrectly) that the time average ofthe total energy, the sum of the kinetic and potentialenergy, has a minimum value between fixed initial and finalevents. To examine the consequences of this guess, let usthink of a ball thrown in a uniform gravitational field,with the two events, pitch and catch, fixed in locationand time (see Fig. 6). How will the ball movebetween these two fixed events? We start by asking whythe baseball does not simply move at constant speed alongthe straight horizontal trajectory B in Fig. 6. Moving frompitch to catch with constant kinetic energy and constant potentialenergy certainly satisfies conservation of energy. But the straight horizontaltrajectory is excluded because of the importance of the averagedpotential energy.

We need to know how the average kineticenergy and average potential energy vary with the trajectory. Webegin with the idealized triangular path T shown in Fig.6. If we assume a fixed time between pitch andcatch and that the speed does not vary wildly alongthe path, the kinetic energy of the particle is approximatelyproportional to the square of the distance covered, that is,proportional to the quantity x + y using the notation in Fig.6. The increase in the kinetic energy over that ofthe straight path is proportional to the square of thedeviation y0, whether that deviation is below or above thehorizontal path. But any incremental deviation from the straight-line pathcan be approximated by a superposition of such triangular incrementsalong the path. As a result, small deviations from thehorizontal path result in an increase in the average kineticenergy approximately proportional to the average of the square ofthe vertical deviation y0.

The average potential energy increases ordecreases for trajectories above or below the horizontal, respectively. Themagnitude of the change in this average is approximately proportionalto the average deviation y0, whether this deviation is smallor large.

We can apply these conclusions to the averageof E = K + U as the path departs from the horizontal. Forpaths above the horizontal, such as C, D, and Ein Fig. 6, the averages of both K and Uincrease with deviation from the horizontal; these increases have nolimit for higher and higher paths. Therefore, no upward trajectoryminimizes the average of K + U. In contrast, for paths thatdeviate downward slightly from the horizontal, the average K initiallyincreases slower than the average U decrease, leading to areduction in their sum. For paths that dip further, however,the increase in the average kinetic energy (related to thesquare of the path length) overwhelms the decrease in theaverage potential energy (which decreases only as y0). So thereis a minimum of the average of K + U for somepath below the horizontal. The path below the horizontal thatsatisfies this minimum is clearly not the trajectory observed fora pitched ball.

Suppose instead we ask how the averageof the difference K–U behaves for paths that deviate fromthe horizontal. By an argument similar to that in thepreceding paragraph, we see that no path below the horizontalcan have a minimum average of the difference. But thereexists a path, such as D, above the horizontal forwhich the average of K–U is a minimum. For thespecial case of vertical launch, students can explore this conclusioninteractively using tutorial software.23

B. Analytic demonstration

We can check thepreceding result analytically in the simplest case we can imagine(see Fig. 7). In a uniform vertical gravitational field amarble of mass m rolls from one horizontal surface toanother via a smooth ramp so narrow that we mayneglect its width. In this system the potential energy changesjust once, halfway between the initial and final positions.

Theworldline of the particle will be bent, corresponding to thereduced speed after the marble mounts the ramp, as shownin Fig. 8. We require that the total travel timefrom position A to position C have a fixed valuettotal and check whether minimizing the time average of thedifference K–U leads to conservation of energy:

where M is given by Eq. (16). We requirethat the value of the action be a minimum withrespect to the choice of the intermediate time t:

Thisresult can be written as

A simple rearrangement shows thatEq. (34) represents conservation of energy. We see that energyconservation has been derived from the more fundamental principle ofleast action. Equally important, the analysis has completely determined theworldline of the marble.

The action Eq. (31) can begeneralized for a potential energy curve consisting of multiple stepsconnected by smooth, narrow transitions, such as the one shownin Fig. 9:

The argument leading to conservation of energy,Eq. (34), applies to every adjacent pair of steps inthe potential energy diagram. The computer can hunt for andfind the minimum value of S directly by varying thevalues of the intermediate times, as shown schematically in Fig.10.

A continuous potential energy curve can be regarded asthe limiting case of that shown in Fig. 9 asthe number of steps increases without limit while the timealong each step becomes an increment t. For the resultingpotential energy curve the general expression for the action Sis

Here L (= K–U for the cases we treat) iscalled the Lagrangian. The principle of least action says thatthe value of the action S is a minimum forthe actual motion of the particle, a condition that leadsnot only to conservation of energy, but also to aunique specification of the entire worldline.

More general forms ofthe principle of least action predict the motion of aparticle in more than one spatial dimension as well asthe time development of systems containing many particles. The principleof least action can even predict the motion of somesystems in which energy is not conserved.24 The Lagrangian Lcan be generalized so that the principle of least actioncan describe relativistic motion7 and can be used to deriveMaxwell's equations, Schroedinger's wave equation, the diffusion equation, geodesic worldlinesin general relativity, and steady electric currents in circuits, amongmany other applications.

We have not provided a proof ofthe principle of least action in Newtonian mechanics. A fundamentalproof rests on nonrelativistic quantum mechanics, for instance that outlinedby Tyc25 which uses the deBroglie relation to show thatthe phase change of a quantum wave along any worldlineis equal to S/, where S is the classical actionalong that worldline. Starting with this result, Feynman and Hibbshave shown26 that the sum-over-all-paths description of quantum motion reducesseamlessly to the classical principle of least action as themasses of particles increase.

Once students have mastered the principleof least action, it is easy to motivate the introductionof nonrelativistic quantum mechanics. Quantum mechanics simply assumes that theelectron explores all the possible worldlines considered in finding theNewtonian worldline of least action.

IX. LAGRANGE'SEQUATIONS

Lagrange's equations are conventionally derived from the principle of leastaction using the calculus of variations.27 The derivation analyzes theworldline as a whole. However, the expression for the actionis a scalar; if the value of the sum isminimum along the entire worldline, then the contribution along eachincremental segment of the worldline also must be a minimum.This simplifying insight, due originally to Euler, allows the derivationof Lagrange's equations using elementary calculus.14 The appendix shows analternative derivation of Lagrange's equation directly from Newton's equations.

ACKNOWLEDGMENTS

APPENDIX: DERIVATION OF LAGRANGE'S EQUATIONS FROM NEWTON'SSECOND LAW

Both Newton's second law and Lagrange's equations can bederived from the more fundamental principle of least action.13,14 Herewe move from one derived formulation to the other, showingthat Lagrange's equation leads to F = ma (and vice versa) forparticle motion in one dimension.28

If we use the definitionof force in Eq. (10), we can write Newton's secondlaw as:

FIGURES

Fig. 1. The potential energy diagram is central to our treatment andrequires a difficult conceptual progression from a ball rolling downa hill pictured in an x-y diagram to a graphicalpoint moving along a horizontal line of constant energy inan energy-position diagram. Making this progression allows the student todescribe qualitatively, but in detail, the motion of a particlein a one-dimensional potential at arbitrary positions such as Athrough F. First citation in article

Fig. 8. Broken worldline of a marble rolling across steps connectedby a narrow ramp. The region on the right isshaded to represent the higher potential energy of the marbleon the second step. First citation in article

Fig. 10. The computer program temporarily fixes the end eventsof a two-segment section, say events B and D, thenvaries the time coordinate of the middle event C tofind the minimum value of the total S, then variesD while C is kept fixed, and so on. (Theend-events A and E remain fixed.) The computer cycles throughthis process repeatedly until the value of S does notchange further because this value has reached a minimum forthe worldline as a whole. The resulting worldline approximates theone taken by the particle. First citation in article